On recognition of $A_6\times A_6$ by the set of conjugacy class sizes

For a finite group $G$ denote by $N(G)$ the set of conjugacy class sizes of $G$. Recently the following question has been asked: Is it true that for each nonabelian finite simple group $S$ and each $n\in\mathbb{N}$, if the set of class sizes of a finite group $G$ with trivial center is the same as the set of class sizes of the direct power $S^n$, then $G\simeq S^n$? In this paper we approach an answer to this question by proving that $A_6\times A_6$ is uniquely determined by $N(A_6\times A_6)$ among finite groups with trivial center.